Call a function \(f\) a \(\mathcal{C}^1\)-function if \(f'\) exists and is continuous. Similarly, call a function \(f\) a \(\mathcal{C}^k\)-function if it has a \(k\)th derivative \(f^{(k)}\) which is continuous. A function is a \(\mathcal{C}^0\)-function if it is continuous. A function is a \(\mathcal{C}^\infty\)-function if all of its derivatives exists and are continuous; alternatively, \(\mathcal{C}^\infty\)-functions are called smooth. As defined, \(\mathcal{C}^k\)-functions are also \(\mathcal{C}^{k-1}\)-functions.

If \(f'\) is a limit then the sum, product, and quotient rules follow.

For example,

\((f + g)' = f' + g'\) because the sum of limits is the limit of the sums.

\((f \cdot g)' = f' \cdot g + f \cdot g'\).

Theorem. There exist functions that are continuous everywhere but are differentiable nowhere.

Theorem (the generalized mean value theorem). If \(f(x)\) and \(g(x)\) are continuous on \([a, b]\) and differentiable on \((a, b)\) then there exists a point \(c \in (a, b)\) such that \((f(b) - f(a))g'(c) = (g(b) - g(a))f'(c)\).